Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $x = \dfrac{5(3t - 1)}{-4} \div \dfrac{3t^2 - t}{t} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{5(3t - 1)}{-4} \times \dfrac{t}{3t^2 - t} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 5(3t - 1) \times t } { -4 \times (3t^2 - t) } $ $ x = \dfrac {t \times 5(3t - 1)} {-4 \times t(3t - 1)} $ $ x = \dfrac{5t(3t - 1)}{-4t(3t - 1)} $ We can cancel the $3t - 1$ so long as $3t - 1 \neq 0$ Therefore $t \neq \dfrac{1}{3}$ $x = \dfrac{5t \cancel{(3t - 1})}{-4t \cancel{(3t - 1)}} = -\dfrac{5t}{4t} = -\dfrac{5}{4} $